On the Geometry of Quantum Spheres and Hyperboloids

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are \(*\)-quantum spaces for the quantum orthogonal group \(\mathcal {O}(SO_q(3))\). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of \(SO_q(3)\). The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules \(\mathcal {E}_n\) of rank 1 and of degree computed to be an even integer \(-2n\). The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra \({\mathcal {U}_{q^{1/2}}(sl_2)}\) which is dual to \(\mathcal {O}(SO_q(3))\).